I
I
R. L. Flurry, Jr. University of New Orleans New Orleans. LA 70122
On the Characters and Representations of C O ~ ~ ~ ~ Point U O UGroups S out by Alvnriilt, (31, it introduces additional ambiguities. Findlv. " ,the method 01' Alvariilo itill contains a certain nmuuot of trial and error. This paper presents a method for carrying out the desired reduction which eliminates all ambiguities and trial-and-error. It works with anv. .DroDerlv . . constructed re~resentationof any of the continltoui point yr,,ups. The methril is hnsed upon the . amtincosine form 1.51 u t rhechar:icteri for the i r r e ~ sufrhe uous groups; consequently, the appendix to the paper presents some relations for handling these characters.
In the past few years several papers have appeared in this Journal concerning the reduction of reducible representations within the continuous (infinite order) point groups (1,2,3). The justification for these papers was that the usual reduction formula 1 ai = -
Zx;* (R)xr(R)
(1)
g R
which is presented in elementary textbooks (4) cannot be used with continuous groups because an infinite summation would be required. (In eqn. (1) a; is the number of times that the irreducible representation (irrep.) i appears in the reducible representation T,g is the order of the group, the R are the operations of the group, x;*(R) is the complex conjugate of the character of R in the irrep. i , and xr(R) is the character of R in the reducible representation.) Schifer and Cyvin (1) point out that the desired reduction for continuous groups can he accomplished by use of the equation x ~ R =) Za; x;(R)
Irreducible Representations for the Continuous Groups 'I'he cosine iurm uf the character for the arbitrary rotation operation, ('(01,uf the irreps. uf the three-dimensional rotae tion grrrup, R(:l),isgi5,en by Hamwmrsh ( 5 1 .T l ~ characters fur the other o~erationsof thr, I ' d I rotation reflecrion group . of the sphere Rk(3), can be derived from thisand the character for the identity, (25 11, by using the character for the inversion, '(25 11, and the identities of Sz and the inversion and of SIand the plane of symmetry. The results for both R(3) and Rh(3) (or O(3)in the mathematical literature) are shown i n 'l'ahle 1. T h r rrpresentations for D,I, and C.. .can he simply derived frurn those of R,,r31 hy recugnizin,: that ihe representatims
+
+
(2)
However, their method involves an amount of trial-and-error if the possible irreps. in the reducible representation are not known in advance. The method of reference (2) attempts to eliminate the need to know the possihle irreps., but, as pointed
Table 1. The Character Table lor the R ( 3 ) and R d 3 ) Polnt Groupsa
+ x 2cosW J
o",
Integer J:
2J+ 1
1
2J+ 1
1
I=,
2J+ I
D:
1
+ g (-1)'2cosl4
(-I)J
I=,
+ zZ C O S I ~
-(2J+
1)
-1
I=,
-
,=,5 ( - 1 ) ' 2 c o s l ~
-(-TIJ
Half-
integer J:
Table 2. The Character Table for the C,, and DmhPolnt Groups* I m -
C-v
E
2q4)
u" -ar
0-h
E
2q4)
-aV
i
--uv
"C, i
2s-4)
I
%E
244)
1
E
244)
-
i
'C2
he gand udistinnion is not applicable in C. . he table includes the aouble-group classes and the double-valued representations. "me h index determines the representation lable lor lntegsr A: II (A = I ) ,A ( h = 2). Q ( A = 3). efc. Fw halt-integer A, the labels are usually expressed as El.
638 / Journal of Chemical Education
2%)
of Rh(3) are reducible in D,h
x J
D $+
ef
x
(integer J )
M'O
J
=+
M
3k:6
(half-integerJ )
(3)
x,
(where p is either g or u). For integer J,€0 corresponds to t o n , €2 to A, etc.. For even J, the irrep, will be and if the representations aregerade. The for odd J it will be converse will hold if they are ungerade. The irreps. for C,, arise from dropping the g and u subscripts. The results are shown in Table 2. €1
x-
x
From the character of C(4) we see immediately that there are 3 x and 2.11 representations. The absence of cos4 in the character for S(-@) indicates that these must be n, and II,. Subtracting these out leaves the following for the representations D-h E C(4) -a, i 2S(-4) "Cz r' 3 3 3 ' 1 1 1 Since ~ ( a , equals ) x(E), these are all while the 1for ~ ( i ) indicates that there are 2 2; and 1 Thus
x+
Reduction of Reducible Representations The key to the reduction of reducible representations is that the representation labels are determined by the multiple of q5 appearing in the character of the arbitrary rotation element C(4) in the two-dimensional rotation groups C,, and D,h and by the highest multiple of q5 for the three-dimensional rotation groups R(3) and Rh(3).Thus, a representation in C ,, which contained a multiple of 2 cos 34 under the C(4) would contain the appropriate number of 4 representations, while a representation in R(3) which contained the same quantity as the highest 1 value would contain the appropriate number of D3 ( F ) representations. In general, for C,, or D-h, the character for a reducible representation under C(4) will have the form, for integer indices. x(c(4))= ao + 2al cos 4 + 2at em 24 t 2a3 cos 34 t(4) This yields the II,A, etc. reduction directly. The ao gives representations, the a1 the number of II the number of representations, etc.. The assignment is completely unamand representations are present, these can biguous. If be separated by the relationship (5) a t = 1/2ix(aJ + a d O;=OO-a$ (6) where an+ is the number of Y+ reuresentations and ah the For the doubc-val"ed representations, there number bf is no constant term and the indices are half-integers. For D-h theg ur u classlficatim can frequently be dvtermined from the wav in which the reducible rrpresentation mas ohtained. (1f theiepesentation was obtained as a product of irreps., all components of the reducible rep. have the same behavior as determined by the g, u multiplication rules.) If this is not the case, it can be unambiguously determined by comparing the sign of the character under S(4) with that under C(4) for each X value. Representations which areg have the characters related by (-1)l for integer X or (-1)("-'/2) for half-integer A, while u representations have the opposite sign relationship. The characters under C(4) for a reducible representation in R(3) or Rh(3) have the form, for integer indices, x(C(&))= (a0 + a , + an + ...ak) + 2(a1+ az + ...ah) cos4 + 2(az + ...aa) cos 24 t ...+ 2 01 cos kb (7) (Agaln, there are no constant terms for half-integer indices.) is ~ccumplishedby svstematicalls subtractma The redurt~ot~ out the representations having the successively highest indices, e.g., D k first, then D(k-'), etc. until the representation is exhausted. This completes the reduction in R(3). The g, u behavior in Rh(3) can almost always be determined from the sc~urceof the representarion by the m ~ ~ l t ~ p l i c a trules. im However, in the cases whvre this is nor so, thesign of the last cosl4 term under S(-@) can again be compared with that under C(q5). The rules are the same as in D-h.
x, x x+ x-
x-.
Examples (1) Normal modes of vibration of acetylene H-C=C-H (point group Dm,,). The reducible vibrational representation is
x+,
xi.
(8) r,,,=z~;+x:+n,+n, (2) Electronic states arising from a a 2configuration of an XY molecule. The Pauli-allowed reducible representation for the singlet state within C,, is C-, E 2 C ( d -au r 4 2 t 2cos 26 0 The character for C(4) tells immediately that there are and one A states. Subtracting out the A representation leaves
2.x
From the character of a, we see that these are thus
x+and I-,
r=x++x-+A (9) (3) Electronicstates arising from a doublet p3 configuration of an atom. The states will all be u since the p orbital transforms as D: within Rh(3); consequently the R(3) group will supply all of the required information. The reducible representation for the Pauli-allowed doublet states in R(3) is R(3) E C i d I' 8 2+4cos4+2coa24 From the last term. we see that this contains D2 one time. Subtracting this out, we have R(3) E C(4) r' 3 1 + 2 cos 4 which is justD1. Thus the doublet states and 2D and 2P. Any other properly constructed representation of the continuous point groups can be reduced in the same manner with equal facility.
Literature Cited (1) Schater, L., and Cyuin,S. J., J.CHEM. EDUC.,4S,295 (1971). (2) Stromrnen. D. P.. and Lippincon,E. H.. J. CHEM. EDUC.,49.341 (19721. (3) Alvsriflo,J.M.. J.CHEM. EDUC.,SJ.307 (19781. (41 Cotton, F.A.,"ChemicdApplications of Grow Theary." Wlley-lntoncienco. N w York. 2nd Ed., 1971. p. 84. (51 Hemermesh, M.."Group Theory? Addison-Wsslq, Reading, Mars.. 1962.p.336. (6) Ford. D. J..J. CHEM.EDUC..49.33611972). (7) Boyle,L. L.,lnl.J. Quantum Chem., 6, 725 (1972). (81 Goninski.O..snd Ohm, Y..Inr. J. QuonlumChem., 2,845(1968).
Appendix The common method of adapting the representations of a group G to S(N) (e.g. of finding Pauli-allowed states) requires the applieation of the character formula (6, 7,8) 1
XIW; [hl =-Xg, e ,
XI,,] (PI
n-- (xr(Ra))u,
(A.1) where xr(R) is the character of the R operation of the r representation af the group being adapted, [A] is the representation of the permutation erouD heine adaoted to. R is the order of the oermutatian group, g, the orde;of the c t h cl& of the permutation group, ~ 1 ~ 1 (P) is the character of the permutation in this class, the a are the cycle lengths of the partition correspondingto permutation P, and u, is the number of cycles of length a in the partition. Applyingeqn. (A.l) to all of the classes of G yields a representation which can be reduced to give the irreducible representationscorresponding to the adaptation. "
-
Volume 56, Number 10, October 1979 1 639
For the three-dimensional rotation group and related groups, the character of the rotation element C(9) is necessaryta determine the allowed reoresentations. However. the character of C(d) is ex~ressed as a funetl'on of either elm, sin 4, dr cos 9 (5).The direct appiication of eqn. (A.1) quickly hecomes tedious due to the necessity of resolving trigonometric identities involving products of different powers of the representation. Simplified formulas for resolving this can be derived for the cos rj construction by considering the exponential construction. We willgive formulasfor [xj(C(9))l", xj(Cm(d))xj(C"(9)),within R(3) as well as (eos m k #) (em n 1 9 ) . This will cover the two- and three-dimensional rotation .. erouos . and the S(N) w i t h N I. 5, which hme at rn8lrt tur, diiierrnf c)clr l m # h m their partitims. Higher .X mn he handled hy chaming the operatiom \Ve will nut derive t h e f m n u l ; i s since t h e algebra is trivial.
640 1 Journal of Chemical Education
R(3): [Xj(C($))]": construct t h e Clehsch-Gordan series for ( D i p a n d t a k e t h e character for C($) f r o m the results.
=
2 2
k=112 1=111
.
(2 cas (mk
- .~
+ nl)9 + 2 cos imk - nll$l
(A.3)
(integer j l
General: (COSm k 4 ) ( e m n 141 = 1/4(2 cos (mk
+ nl)9 + 2 coslrnk - nll91 (A.41
